Gyroscope utilizing torsional springs and optical sensing

ABSTRACT

A gyroscope and a method of detecting rotation are provided. The gyroscope includes a structure configured to be driven to move about a drive axis. The structure is further configured to move about a sense axis in response to a Coriolis force generated by rotation of the structure about a rotational axis while moving about the drive axis. The structure further includes at least one first torsional spring extending generally along the drive axis and at least one second torsional spring extending generally along the sense axis. The gyroscope further includes an optical sensor system configured to optically measure movement of the structure about the sense axis.

CLAIM OF PRIORITY

The present application is a continuation of U.S. patent applicationSer. No. 12/770,545, filed Apr. 29, 2010 and incorporated in itsentirety by reference herein, which claims the benefit of priority toU.S. Provisional Appl. No. 61/174,969, filed on May 1, 2009, andincorporated in its entirety by reference herein.

BACKGROUND

1. Field of the Invention

This application relates generally to gyroscopes, and more specifically,to gyroscopes utilizing optical sensing.

2. Description of the Related Art

Known optical fiber gyroscopes (e.g., fiber-optic gyroscopes or fiberring gyroscopes) do not have mechanical parts and are based on theSagnac effect. While miniature mechanical gyroscopes are known (See,e.g., J. J. Bernstein, U.S. Pat. No. 5,203,208 and T. K. Tang et al.,U.S. Pat. No. 5,894,090), conventional miniature mechanical gyroscopesare generally based on microelectromechanical system (MEMS) technology,and the rotation applied to the gyroscope is sensed using electrostaticsor some form of magnetic sensing. The rotation sensitivity ofconventional MEMS-based gyroscopes is limited, and several orders ofmagnitude worse than Sagnac-based optical gyroscopes.

For example, the performance of conventional MEMS-based gyroscopes isusually limited by the electronic noise, which is fairly high.Therefore, these existing gyroscopes must be operated at a mechanicalresonance frequency of the structure (e.g., of the two or moreoscillating plates) in order to enhance the signal resulting from theapplied rotation. In order for the two plates of such conventionaldevices to be operated on resonance, they must exhibit at least one setof identical resonance frequencies. Achieving identical resonancefrequencies requires very accurate tuning of the structural parametersduring fabrication, which is often limited by fabrication tolerances. Toachieve good sensitivity with such configurations, the mechanicalquality factor is designed to be very large, making it very hard todesign a structure in which the mechanical drive and sense frequenciesmatch. A high quality factor also reduces the measurement bandwidth,i.e., the dynamic range of the sensor, since the bandwidth scales withthe inverse of the quality factor.

As a result of these complexities, and of the fairly high electronicnoise, current MEMS gyroscopes exhibit a relatively low sensitivity. Atypical good MEMS gyroscope usually can detect in the range of 0.1 to 1deg/s. See, e.g., C. Acar and A. Shkel, MEMS vibratory gyroscopes:structural approaches to improve robustness, Springer (2008). A minimumdetectable rotation rate of 0.05 deg/s has also been reported. See H.Xie and G. K. Fedder, “Integrated microelectromechanical gyroscopes,” J.Aerospace Eng. Vol. 16, p. 65 (2003). There are a few reports of MEMSgyroscopes with much better sensitivities (˜10 deg/h) (see, e.g., Acarand Shkel), but they operate in vacuum and have very tightly matcheddrive and sense modes. See, e.g., T. K. Tang, R. C. Gutierrez, J. Z.Wilcox, C. Stell, V. Vorperian, M. Dickerson, B. Goldstein, J. L.Savino, W. J. Li, R. J. Calvet, I. Charkaborty, R. K., Bartman, and W.J. Kaiser, “Silicon bulk micromachined vibratory gyroscope formicrospacecraft, “Proc. SPIE Vol. 2810, p. 101 (1996). Suchconfigurations may be difficult to reproduce on a large scale and at alow cost.

SUMMARY

In certain embodiments, a gyroscope is provided. The gyroscope comprisesa structure configured to be driven to move about a drive axis. Thestructure is further configured to move about a sense axis in responseto a Coriolis force generated by rotation of the structure about arotational axis while moving about the drive axis. The gyroscope furthercomprises an optical sensor system configured to optically measuremovement of the structure about the sense axis. In certain embodiments,the gyroscope is a microelectromechanical system (MEMS) gyroscope.

In certain embodiments, a method of detecting rotation is provided. Themethod comprises providing a structure configured to be driven to moveabout a drive axis and to move about a sense axis in response to aCoriolis force generated by rotation of the structure about a rotationalaxis while moving about the drive axis. The method further comprisesdriving the structure to move about the drive axis. The method furthercomprises rotating the structure about the rotational axis while thestructure moves about the drive axis. The method further comprisesoptically measuring movement of the structure about the sense axis.

In certain embodiments, optically measuring movement of the structurecomprises irradiating at least a portion of the structure withelectromagnetic radiation and receiving reflected electromagneticradiation from the portion of the structure. In certain embodiments,optically measuring movement of the structure further comprisesdetecting at least a portion of the received reflected electromagneticradiation and generating one or more signals in response to the detectedportion of the received reflected electromagnetic radiation.

In certain embodiments, the method further comprises centering the postportion on the structure. In certain such embodiments, centering thepost portion comprises placing a sensor on the sense axis; measuring afirst noise spectrum having a first peak at a resonance frequency of asense mode; placing the sensor on the drive axis; measuring a secondnoise spectrum having second peak at a resonance frequency of a drivemode; and determining a position of the post portion on the structurewhere the first and second peaks are reduced.

For the gyroscope and/or the method of detecting rotation, the structureof certain embodiments comprises a generally planar portion, at leastone first torsional spring extending generally along the drive axis andoperationally coupling the generally planar portion to a supportstructure, and at least one second torsional spring extending generallyalong the sense axis and operationally coupling the generally planarportion to the support structure. The generally planar portion ofcertain embodiments comprises at least two drive arms extending inopposite directions from one another generally along the sense axis,wherein the at least second drive torsional spring operationally couplesthe at least two drive arms to the support structure. The generallyplanar portion of certain embodiments comprises at least two sense armsextending in opposite directions from one another generally along thedrive axis, wherein the at least one first torsional springoperationally couples the at least two sense arms to the supportstructure.

The structure of certain embodiments comprises a post portion extendinggenerally perpendicularly away from the generally planar portion. Therotational axis of certain embodiments is substantially perpendicular toat least one of the drive axis and the sense axis.

The optical sensor system of certain embodiments comprises one or moreoptical fibers configured to irradiate at least a portion of thestructure with electromagnetic radiation and to receive reflectedelectromagnetic radiation from the portion of the structure. In certainembodiments, the one or more optical fibers and the portion of thestructure form at least one Fabry-Perot cavity therebetween. The opticalsensor system of certain embodiments further comprises one or moreoptical detectors in optical communication with the one or more opticalfibers, with the one or more optical detectors configured to receiveelectromagnetic radiation reflected from the portion of the structureand transmitted by the one or more optical fibers and to generate one ormore signals in response to the received electromagnetic radiation. Theportion of the structure of certain embodiments comprises one or morephotonic-crystal structures.

In certain embodiments, the gyroscope further comprises a drive systemconfigured to drive the structure to oscillate about the drive axis. Thedrive system of certain embodiments comprises one or more optical fibersconfigured to irradiate at least a portion of the structure withelectromagnetic radiation having sufficient radiation pressure to drivethe structure to oscillate about the drive axis. In certain embodiments,the one or more optical fibers and the portion of the structure form atleast one Fabry-Perot cavity therebetween. In certain other embodiments,the drive system comprises one or more electrodes configured to applysufficient electrostatic force on at least a portion of the structure todrive the structure to oscillate about the drive axis.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 a and 1 b schematically illustrate example gyroscopes inaccordance with certain embodiments described herein.

FIG. 2 shows a simplified drawing of an example gyroscope in accordancewith certain embodiments described herein.

FIG. 3 shows a drawing of an example gyroscope in accordance withcertain embodiments described herein.

FIG. 4 shows a drawing of an example gyroscope in accordance withcertain embodiments described herein.

FIG. 5 a schematically illustrates an example sensing mechanism usingfiber Fabry-Perot interferometers in accordance with certain embodimentsdescribed herein.

FIG. 5 b schematically illustrates an example driving mechanism usingradiation pressure in accordance with certain embodiments describedherein.

FIG. 6 schematically illustrates a closed-loop operation of a gyroscopein accordance with certain embodiments described herein.

FIG. 7 shows the MEMS structure of H. Ra, W. Piyawattanametha, Y.Taguchi, D. Lee, M. J. Mandella, and O. Solgaard, “Two-dimensional MEMSscanner for dual-axes confocal microscopy,” J. Microelectromech. Syst.16, 969 (2007).

FIGS. 8-9 are flowcharts of example methods for detecting rotation inaccordance with certain embodiments described herein.

DETAILED DESCRIPTION

Certain embodiments described herein include a mechanical structure andare based on the Coriolis effect. The device in accordance with certainembodiments described herein is based on a miniature Foucault pendulum.The set of limitations on the performance of certain embodimentsdescribed herein is different from the limitations of Sagnac-basedoptical gyroscopes. Therefore, a gyroscope in accordance with certainembodiments described herein is not limited by the problems limitingknown Sagnac-based fiber-optic gyroscopes. In certain embodiments, thegyroscope can be made very small in size (e.g., within a few millimetersin every dimension). Therefore, certain such embodiments providesubstantial reductions in size which can be advantageous for certainapplications. Also, certain embodiments described herein provide thepossibility of mass production, hence significant reduction in cost.

In addition, in contrast to conventional MEMS-based gyroscopes, incertain embodiments described herein, rotation is sensed optically,thereby providing an advantage in the noise performance of thegyroscope, which translates in a far superior rotation sensitivity. Incertain embodiments described herein, the gyroscope is also drivenoptically, thereby reducing the electromagnetic interference due to theelectrical drive normally used in MEMS gyroscopes.

Employing optics in sensing instead of electrostatics or some form ofmagnetic sensing in certain embodiments described herein canadvantageously reduce the limiting noise to the mechanical thermal noiseof the structure. This noise has the same frequency dependence as doesthe signal. Therefore, at a resonance frequency of the structure, thenoise is high like the signal, and at non-resonant frequencies, thenoise is low like the signal. Therefore, for optical sensing, thesignal-to-noise ratio (SNR) is not improved when the sensing is done ata mechanical resonance frequency (e.g., 100 Hz to 10 kHz, sensitivitiesare typically higher in MEMS gyroscopes at lower frequencies).Conversely, for optical sensing, the signal-to-noise ratio (SNR) is notdegraded when the sensing is done away from a mechanical resonancefrequency. In certain embodiments, sensing can be performed at anon-resonant frequency, thereby improving the bandwidth significantlywithout compromising the SNR. Also, by operating at non-resonantfrequencies in certain embodiments, the settling time of the gyroscopecan be reduced, so that faster rotation sensing can be performed. Forthe example gyroscope discussed below, certain embodiments describedherein can lead to a minimum detectable rotation of about 5 deg/h (witha 1-Hz detection bandwidth). With straightforward improvements alsosuggested below, this figure can be improved in certain embodiments(e.g., 1 deg/h for the same 1-Hz bandwidth). A fiber-optic gyroscope(FOG) in accordance with certain embodiments described herein can easilydetect 1/1000th of the Earth's rotation rate, or 0.015 deg/h, and evenbetter if the long-term stability requirement is relaxed.

In recent years there has been a surge in the commercial development ofgimbaled, large amplitude, two-axis scanning MEMS mirrors. These mirrorsare intended to be used in projectors (called pico-projectors) inportable devices such as cellular phones. There are several companiesthat produce these mirrors, such as Hiperscan GmbH, Microvision Inc.,and Electro-Optical Products Corp. By modifying these commerciallyavailable micro-mirrors with certain embodiments described herein, it ispossible to build optomechanical fiber gyroscopes in a cheaper way.

FIG. 1 a schematically illustrates an example gyroscope 10 in accordancewith certain embodiments described herein. The gyroscope 10 of certainembodiments comprises a structure 100 configured to be driven to moveabout a first axis, e.g., drive axis 102. The structure 100 is furtherconfigured to move about a second axis, e.g, sense axis 104, in responseto a Coriolis force generated by rotation of the structure 100 about arotational axis z while moving (e.g., vibrating) about the drive axis102. The gyroscope 10 of certain embodiments further comprises anoptical sensor system 200 configured to optically measure movement(e.g., vibration) of the structure 100 about the sense axis 104.

In certain embodiments, the gyroscope 10 is a microelectromechanicalsystem (MEMS) gyroscope. In certain embodiments, the gyroscope 10comprises one or more components which are micromachined (e.g., from asubstrate). The gyroscope 10 can comprise one or more components whichare made of silicon, polysilicon, silica, or quartz, or combinationsthereof. Certain embodiments can include silicon-nitride layers forstress compensation. Also, metal layers (e.g., polished or depositedsurfaces comprising gold, silver, aluminum, chrome, titanium, platinum),conventional multilayer dielectric coatings, or photonic crystals can beused in certain embodiments described herein to enhance opticalreflections when an optical sensing method is employed, as describedmore fully below.

In certain embodiments, the structure 100 comprises a generally planarportion 105. For example, FIG. 2 is a simplified drawing of an examplegyroscope 10 in accordance with certain embodiments described herein. Incertain embodiments, the generally planar portion 105 comprises a baseplate 107. The base plate 107 of certain embodiments comprises a singlegenerally planar plate having four generally perpendicular arms, whilecertain other embodiments, the base plate 107 comprises two or moreplates mechanically coupled to one another. In still other certainembodiments, the base plate 107 has another shape (e.g., square,circular, polygonal, irregular), with or without arms.

As shown in FIG. 2, the generally planar portion 105 of the structure100 in certain embodiments comprises at least two drive arms 150, 160extending in opposite directions from one another generally along thesense axis 104. For example, the drive arms 150, 160 can comprisesilicon and have a length of about 1400 microns, in a range betweenabout 500 microns and 2500 microns, or in a range between about 1000microns and about 2000 microns.

In certain embodiments, the generally planar portion 105 of thestructure 100 comprises at least two sense arms 180, 190 extending inopposite directions from one another generally along the drive axis 102.For example, the sense arms 180, 190 can comprise silicon and have alength of about 1400 microns, in a range between about 500 microns and2500 microns, or in a range between about 1000 microns and about 2000microns.

The structure 100 further comprises at least one first torsional spring110, as schematically illustrated in FIG. 1 b, extending generally alongthe drive axis 102 and operationally coupling the generally planarportion 105 to a support structure 300. The structure 100 also comprisesat least one second torsional spring 120 extending generally along thesense axis 104 and operationally coupling the generally planar portion105 to the support structure 300. For example in FIG. 2, the structure100 comprises two second torsional springs 120 operationally coupled tothe two drive arms 150, 160 and to the support structure 300 (notshown), and further comprise two first torsional springs 110operationally coupled to the two sense arms 180, 190 and to the supportstructure 300 (not shown).

In certain embodiments, the torsional springs 110, 120 may be one of avariety of torsional springs well known in the art, although other typesof springs well known in the art or yet to be devised may be used aswell. In certain embodiments, the torsional springs comprise the samematerial as does the generally planar portion 105 of the structure 100.For example, the at least one first torsional spring 110 can comprisesilicon, have a length in a range between about 100 microns and about500 microns (e.g., about 350 microns), a width in a range between 2microns and 15 microns (e.g., about 8 microns), and a thickness in arange between 10 microns and 50 microns (e.g., about 30 microns), andthe at least one second torsional spring 120 can comprise silicon, havea length in a range between about 100 microns and about 500 microns(e.g., about 350 microns), a width in a range between 2 microns and 15microns (e.g., about 8 microns), and a thickness in a range between 20microns and 100 microns (e.g., about 60 microns). In certainembodiments, the springs are generally rectangular-shaped, but othershapes (e.g., serpentine-shaped) are also compatible with variousembodiments described herein, depending on the space constraints andstiffness requirements.

In certain embodiments, the drive axis 102 is substantially planar withand substantially perpendicular to the sense axis 104. While having anangle of about 90° between the drive axis 102 and the sense axis 104 isadvantageous, in certain other embodiments, the drive axis 102 and thesense axis 104 are substantially planar with one another but have anangle different from 90° between the drive axis 102 and the sense axis104. In certain embodiments, the drive axis 102 and the sense axis 104are not co-planar with one another. The structure 100 of certainembodiments can vibrate or oscillate around each of the two axes 102 and104.

As schematically illustrated in FIG. 1 b, certain embodiments furthercomprise a drive system 400 configured to drive the structure 100 tovibrate or oscillate at a certain frequency ω around or about the driveaxis 102 with an angular deflection θ. This motion can be driven eitheroptically (e.g., employing radiation pressure) or electrostatically(e.g., using MEMS electrodes).

In certain embodiments that are driven optically, the drive system 400can comprise one or more optical fibers optically coupled tocorresponding portions of the structure 100. For example, FIG. 3 is asimplified drawing of certain embodiments comprising a drive system 400configured to drive the structure 100 optically. The drive system 400 ofFIG. 3 comprises a pair of optical fibers 410, 420 (and denoted byarrows with vertical hatchings), each of which is optically coupled to acorresponding portion of the structure 100 to form two Fabry-Perotinterferometers 440, 450. In certain such embodiments, at least oneFabry-Perot interferometer 440, 450 is formed between a portion 430 ofthe structure 100 (e.g., corresponding portions of the drive arms 150,160) and one or more optical fibers 410, 420. The one or more opticalfibers 410, 420 are configured to irradiate at least the portion 430 ofthe structure 100 (e.g., corresponding portions near the tips of thedrive arms 150, 160) with electromagnetic radiation having sufficientradiation pressure to drive the structure 100 to move (e.g., oscillate)about the drive axis 102 (e.g., with an amplitude large enough toachieve a desired level of sensitivity of at least 0.0015 degree/hour,or across a range between 0.0015 degree/hour and 15 degrees/hour, oracross a range between 0.0015 degree/hour and 1.296×10⁷ degrees/hour).The radiation pressure caused by the photon flux inside each of thesetwo Fabry-Perot interferometers 440, 450 drives the vibrations oroscillations of the structure 100 around the drive axis 102. In certainother embodiments, the one or more optical fibers do not formFabry-Perot cavities with the structure 100, but do provide sufficientradiation pressure to bring the movement of the structure 100 about thedrive axis 102.

In certain embodiments which utilize a pair of optical fibers 410, 420on opposite arms 150, 160 of the structure 100 to drive the vibrationsor oscillations around the drive axis 102 (e.g., FIG. 3), the wavelengthof the laser light used to drive the radiation pressure Fabry-Perotinterferometers 440, 450 is set to one of the optical resonancewavelengths of the fiber Fabry-Perot interferometers 440, 450. Incertain such embodiments, the light used in the two Fabry-Perotinterferometers 440, 450 to drive the structure 100 can be modulated inopposite phases with one another. This configuration can increase thetorque about the drive axis 102, which can yield larger driveamplitudes.

In certain embodiments, the radiation pressure is produced by launchinglight of sufficient intensity through an optical fiber directly near atip of the drive-axis arms without using a Fabry-Perot cavity to enhancethe optical flux. This approach typically utilizes more power than theFabry-Perot approach. However, because it does not use a Fabry-Perotcavity, the stability of the device is improved, a benefit that mayoutweigh the increased power requirement. Which of the two approaches isto be selected depends to a large degree on the trade-off between ahigher power requirement (which likely translates into a higher cost)and higher stability (which translates into a poorer long-term stabilityand/or more complex engineering to improve the stability). Thistrade-off can be easily analyzed theoretically using basic concepts inoptics. In addition, in certain embodiments in which multiple portionsof the structure 100 on generally opposite sections of the structure 100(e.g., either on generally opposite top and bottom surfaces or onportions located on generally opposite sides in relation to thecenter-of-gravity or pivot point of the structure 100) are irradiated bylight emitted by multiple optical fibers, the light emitted by theseoptical fibers used to drive the structure 100 can be also modulated(e.g., in opposite phases with one another), which can have similarbenefits as pointed out above.

In certain embodiments, the drive system 400 comprises at least oneconventional fiber (e.g., a single-mode fiber such as the SMF-28®optical fiber available from Corning, Inc. of Corning, N.Y.). In certainembodiments, the one or more fibers of the drive system 400 can beeither a single-mode fiber or a multimode fiber (e.g., INFINICOR® fiberavailable from Corning, Inc. of Corning, N.Y.). In certain embodiments,the one or more fibers of the drive system 400 can advantageously be apolarization-maintaining fiber, or a polarizing fiber, to excite asingle and stable state of polarization in the Fabry-Perotinterferometers during operation. In certain such embodiments,fluctuations in the response of the FP sensor caused by birefringence inthe FP interferometers is advantageously removed, for example, due toany dependence of the reflectivities on polarization (although suchdependencies are small in certain embodiments). The spacing between thefiber and the plate (or equivalently the cavity length for a FPinterferometer) (e.g., in a range between about 10 microns to about 1millimeter) is not critical for certain embodiments (e.g., in which a FPinterferometer is not used for the irradiation). In certain embodiments,the irradiated area is the portion or surface of the structure 100underneath each fiber tip on the drive arms 150, 160. In certain suchembodiments in which the light impinges generally perpendicularly to agenerally flat surface of the drive arms 150, 160, the forces producedby this irradiation are generally perpendicular to the surface of thedrive arms 150, 160.

In certain embodiments, the drive system 400 is configured to drive thestructure 100 electrostatically, using MEMS electrodes, withoututilizing the fiber Fabry-Perot interferometers described above. Forexample, as shown in FIG. 4, the drive system 400 can comprise one ormore electrodes 460, 470 configured to apply sufficient electrostaticforce on at least a portion 480 of the structure 100 to drive thestructure 100 to oscillate about the drive axis 102 (e.g., with anamplitude large enough to achieve a desired level of sensitivity of atleast 0.0015 degree/hour, or across a range between 0.0015 degree/hourand 15 degrees/hour, or across a range between 0.0015 degree/hour and1.296×10⁷ degrees/hour). In certain embodiments, the one or moreelectrodes 460, 470 may be one of a variety of MEMS electrodes wellknown in the art, although other types of electrodes well known in theart or yet to be devised may be used as well. For example, in certainembodiments, the electrodes 460, 470 can comprise parallel-plateelectrodes or comb-drive electrodes. The electrodes 460, 470 can beeither at generally opposite top and bottom surfaces of the structure100 or at portions of the structure 100 located on generally oppositesides in relation to the center-of-gravity or pivot point of thestructure 100. The electrostatic force is attractive, such that thestructure 100 is attracted towards the electrodes 460, 470 when voltageis applied. In certain other embodiments, one or more electromagnets canbe used to provide the driving force on the structure 100. For example,in certain embodiments, a combination of a permanent magnet and anelectromagnet, or a combination of two electromagnets, can be used. Incertain such embodiments, at least one magnet is positioned on thestructure 100 and at least one magnet is spaced from the structure 100,similar to the electrode configuration, to produce magnetic forces todrive the structure 100. Example configurations compatible with certainembodiments described herein to generate the driving force are describedby J. Korvink and O. Paul (eds.), “MEMS—A Practical Guide to Design,Analysis and Applications,” William Andrew Publishing, Norwich, 2006,pp. 345-402, and references therein.

In certain embodiments, the structure 100 comprises a post portion 310.The post portion 310 of certain embodiments is mechanically coupled toand extending generally perpendicular away from the generally planarportion 105 of the structure 100. For example, in certain embodiments,as schematically illustrated by FIGS. 2, 3, and 4, the post portion 310extends generally along the rotational axis z which is substantiallyperpendicular to the drive axis 102 and the sense axis 104. While havingan angle of about 90° between the rotational axis z and one or both ofthe drive axis 102 and the sense axis 104 is advantageous, in certainother embodiments, the rotational axis z has an angle different from 90°between it and one or both of the drive axis 102 and the sense axis 104.In certain such embodiments, such a tilt angle may induce asymmetry inthe oscillation which may not be advantageous, or may induce wobblingwhich may also not be advantageous.

In certain embodiments, the post portion 310 is located at the center(e.g., the center of mass) of the generally planar portion 105. Incertain embodiments in which the post portion 310 comprises multipleportions, these multiple portions can be spaced away from the center ofthe generally planar portion 105 but are positioned generallysymmetrically about the center (e.g., such that wobbling of thestructure 100 is avoided, minimized, or reduced, or such thatsensitivity is not degraded). When the gyroscope 10 experiences arotation around the z-axis (e.g., the axis along the post portion 310),the Coriolis effect will induce a force 1000 in a direction (shown bythe dashed, double-headed arrow in FIGS. 2 and 3) perpendicular to thedirection (shown by the solid, double-headed arrow in FIGS. 2 and 3) ofthe motion of the post portion about the drive axis 102. Therefore, thepost portion 310 will vibrate or oscillate about or around the senseaxis 104 with an angular deflection φ at the drive frequency ω.

In certain embodiments, the post portion 310 can be made of the samematerials as that of the structure 100. For example, the post portion310 can be made of a generally rigid material, including but not limitedto, silicon, polysilicon, silica, quartz, iron, brass, glass, hardplastic, or Teflon. For example, in certain embodiments, the postportion 310 comprises a length (e.g., 10-20 millimeters) of conventionaloptical fiber, stripped of its jacket. In certain embodiments, the postportion 310 is shaped (e.g., lollipop-shaped) such that its mass isconcentrated towards the top of the post portion 310 rather than thebottom of the post portion 310 where it is mechanically coupled to theplate portion 110 of the structure. Certain such embodiments can providea better performance than post portions 310 with a uniformly distributedmass.

In certain embodiments, the vibrations or oscillations about the senseaxis 104 are measured optically using an optical sensor system 200configured to provide information about the rotation rate of thegyroscope 10 about the z-axis. FIG. 3 shows an example gyroscope 10driven optically and measured optically, while FIG. 4 shows an examplegyroscope 10 driven electrostatically and measured optically. In certainembodiments, the optical sensor system 200 comprises one or more fiberFabry-Perot interferometers (see, e.g., U.S. Pat. No. 7,526,148, U.S.Pat. No. 7,630,589, and U.S. Pat. Appl. Publ. No. 2008/0034866A1, eachof which is incorporated in its entirety by reference herein).

For example, in certain embodiments, the optical sensor system 200comprises one or more optical fibers. In each of FIGS. 3 and 4, theexample gyroscope 10 comprises a pair of optical fibers 610, 620(denoted by arrows with diagonal hatchings in FIG. 3) with each opticalfiber 610, 620 forming a Fabry-Perot interferometer with a correspondingportion 630 of the structure 100 (e.g., portions of the two sensing arms180, 190 of the generally planar portion 105 of the structure 100) tomeasure the sense-axis 104 oscillations in accordance with certainembodiments described herein. Each of the optical fibers 610, 620 isconfigured to irradiate at least a portion 630 of the structure 100 withelectromagnetic radiation and to receive reflected electromagneticradiation from the portion 630 of the structure 100. In certain suchembodiments, a Fabry-Perot cavity is formed between the portion 630 ofthe structure 100 and the one or more optical fibers 610, 620. Forexample, the cavity length can be in a range between about 10 micronsand about 1 millimeter.

In certain embodiments, the sensor system 200 comprises at least oneconventional fiber (e.g., a single-mode fiber such as the SMF-28®optical fiber available from Corning, Inc. of Corning, N.Y.). In certainembodiments, the one or more fibers of the sensor system 200 can beeither a single-mode fiber or a multimode fiber (e.g., INFINICOR® fiberavailable from Corning, Inc. of Corning, N.Y.). While multimode fibersprovide less sensitivity, certain such embodiments can be useful inwhich sensitivity can be sacrificed over ease of assembly and lessstringent fiber-spacing parameters. In certain embodiments, the one ormore fibers of the sensor system 200 can advantageously be apolarization-maintaining fiber, or a polarizing fiber, to excite asingle and stable state of polarization in the Fabry-Perotinterferometers during operation. In certain such embodiments,fluctuations in the response of the FP sensor caused by birefringence inthe FP interferometers is advantageously removed, for example, due toany dependence of the reflectivities on polarization (although suchdependencies are small in certain embodiments). Spacing between thefiber and the plate is not critical for certain embodiments in which aFP interferometer is not used for the irradiation. In certainembodiments, the irradiated area is the portion or surface of thestructure 100 underneath each fiber tip on the sense arms 180, 190. Thespacings or cavity lengths for the FP interferometers on the sensingarms 180, 190 is somewhat more important than are the spacings or cavitylengths for the FP interferometers on the drive arms 150, 160. For thesensing FP interferometers, as the spacing is reduced, the light thatpropagates through the FP interferometer and is reflected back to thefiber has a shorter distance to travel before being recoupled into thefiber, so the light does not diffract as much. As a result, therecoupling loss is smaller, which translates into a higher finesse forthe FP interferometers, and hence a greater sensitivity. There is ofcourse diminishing returns when the spacing becomes extremely small. Thespacing at which this diminishing return starts becoming apparentdepends on the loss of the FP interferometer: the lossier it is, theless important diffraction loss is in relative terms, and the larger thespacing can be without compromising the finesse and sensitivity. Thistrade-off is discussed in a paper by Onur Kilic, Michel J. F. Digonnet,Gordon S. Kino, and Olav Solgaard, “Asymmetrical spectral response infiber Fabry-Perot interferometers,” IEEE J. Lightwave Technol. Vol. 27,No. 24, 5648-5656 (December 2009), which is incorporated in its entiretyby reference herein.

The sensor system 200 of certain embodiments can comprise one or morecomponents made of silicon, polysilicon, silica, or quartz and cancomprise micromachined portions. In certain embodiments, the sensorsystem 200 can include silicon-nitride layers for stress compensation.Also, the sensor system 200 can include metal layers (e.g., polished ordeposited surfaces comprising gold, silver, aluminum, chrome, titanium,platinum), conventional multilayer dielectric coatings, or photoniccrystals in certain embodiments described herein to enhance the opticalreflectivity of the portions of the structure 100 which are opticallymonitored (e.g., the portions 630 optically coupled to the one or moreoptical fibers).

FIG. 5 a schematically illustrates an example optical sensor system 200using fiber Fabry-Perot interferometers as optical sensors in accordancewith certain embodiments described herein. In certain embodiments, theoptical sensor system 200 comprises one or more optical detectors 250,260 in optical communication with one or more optical fibers 610, 620which are in optical communication with corresponding portions 630 ofthe structure 100. In certain embodiments, the one or more opticaldetectors 250, 260 may be one of a variety of photodetectors well knownin the art, although detectors yet to be devised may be used as well.For example, low-noise detectors such as ones made from small-areaindium-gallium-arsenide PIN photodiodes can be used. Fiber-coupleddetectors are preferable over free-space detectors for practicalreasons, although both are compatible with certain embodiments describedherein.

The one or more optical detectors 250, 260 are configured to receiveelectromagnetic radiation reflected from the portion 630 of thestructure 100. In certain embodiments, the electromagnetic radiationimpinging on the portion 630 is transmitted by the one or more opticalfibers 610, 620 and at least some of the reflected portion of thiselectromagnetic radiation is received by the one or more optical fibers610, 620. The one or more optical detectors 250, 260 are configured toreceive at least a portion of the reflected portion of theelectromagnetic radiation received by the one or more optical fibers610, 620 and to generate one or more signals in response to the receivedreflected electromagnetic radiation from the portion 630 of thestructure 100.

In FIG. 5 a, the one or more optical fibers 610, 620 and thecorresponding portions 630 of the structure 100 form two fiberFabry-Perot interferometers 710, 720 and a laser 800 provideselectromagnetic radiation to the two Fabry-Perot interferometers 710,720 at a wavelength that provides high displacement sensitivity in thefiber Fabry-Perot interferometers 710, 720. Specifically, the wavelengthof the light in certain embodiments is selected to fall on or very neara steepest slope of the reflection spectrum of the FP interferometer formaximum sensitivity. In certain embodiments, the laser 800 may be one ofa variety of lasers well known in the art (e.g., low-noise lasersources). For example, in certain embodiments, the laser 800 is anarrowband laser. Lasers yet to be devised may be used as well.Fiber-coupled lasers are preferable over free-space lasers for practicalreasons, although both are compatible with certain embodiments describedherein.

In certain embodiments, it is advantageous to operate at a wavelengtharound 1.5 μm, where the loss of the optical fibers is smaller, butother wavelengths are also compatible with certain embodiments describedherein. In certain embodiments, a shorter wavelength is advantageouslyused (e.g., below approximately 1.1 μm) which allows use of a silicondetector, which typically has a lower noise than detectors used atlonger wavelengths. The wavelength can be selected in accordance withthe teachings of O. Kilic, “Fiber based photonic-crystal acousticsensor,” Ph.D. Thesis, Stanford University (2008) and O. Kilic, M.Digonnet, G. Kino, and O. Solgaard, “External fibre Fabry-Perot acousticsensor based on a photonic-crystal mirror,” Meas. Sci. Technol. 18, 3049(2007), which are both incorporated in their entireties by referenceherein.

In certain embodiments, the laser wavelengths and the size of the one ormore Fabry-Perot cavities of the one or more fiber Fabry-Perotinterferometers 710, 720 (e.g., the spacings between the ends of the oneor more optical fibers 610, 620 and the portions 630 of the structure100) are selected to provide a desired level of thermal stability. Incertain embodiments, these physical parameters, as well as others, ofthe optical sensor system 200 are selected in accordance with theteachings of O. Kilic, “Fiber based photonic-crystal acoustic sensor,”Ph.D. Thesis, Stanford University (2008) and O. Kilic, M. Digonnet, G.Kino, and O. Solgaard, “External fibre Fabry-Perot acoustic sensor basedon a photonic-crystal mirror,” Meas. Sci. Technol. 18, 3049 (2007),which are both incorporated in their entireties by reference herein. Incertain embodiments, the wavelength of the light used to probe these twoFabry-Perot interferometers 710, 720 is selected to fall on the steepestportion of the resonance of the Fabry-Perot interferometers 710, 720 tomaximize the Fabry-Perot interferometers' sensitivity to smalldisplacements. See, e.g., O. Kilic, “Fiber based photonic-crystalacoustic sensor,” Ph.D. Thesis, Stanford University (2008). For example,the laser can be a telecom-type laser with a typical wavelength of 1550nanometers, the cavity spacing of the FP interferometers is betweenabout 10 microns and 150 microns, and the displacement sensitivity isabout 10⁻⁵ nanometers in a 1-Hz measurement bandwidth.

In certain embodiments, as schematically illustrated by FIG. 5 a, thelaser light from the laser 800 is coupled into a fiber 660 that isoptically coupled to a 3-dB coupler 900 so that the optical power of thelight is substantially equally separated into two arms 910, 920. In eacharm 910, 920, the optical power is transmitted to a second 3-dB coupler960, 970 and is transmitted to corresponding fiber Fabry-Perotinterferometer 710, 720. The two Fabry-Perot interferometers 710, 720 incertain embodiments are on opposite sides of the sense axis 104 andoscillate or vibrate in opposite phases with one another (e.g., when theFabry-Perot cavity spacing of one of the interferometers increases, theFabry-Perot cavity spacing of the other interferometer decreases). Thesignal returning from each Fabry-Perot interferometer is transmitted tothe corresponding second 3-dB coupler 960, 970 and at least a portion ofthe signal is sent to the corresponding optical detector 250, 260. Incertain embodiments, the difference signal corresponding to a differencebetween the signals of the two optical detectors 250, 260 is used toobtain the rotation rate of the structure 100 about the sense axis 104.In certain embodiments, the signals from the two optical detectors 250,260 can also be added to generate a sum signal which is used in certainembodiments to provide information regarding the acceleration of thegyroscope 10 along the z-axis. Other types of couplers 960, 970 are alsocompatible with certain embodiments described herein (e.g., 10%-90%couplers or optical circulators).

FIG. 5 b illustrates an example driving mechanism using radiationpressure in accordance with certain embodiments described herein. FIG. 5b shows that the structure 100 can be oscillated or vibrated around thedrive axis 102 by modulating the laser 800 at the fundamental torsionalresonance frequency of the drive axis 102. In certain embodiments, thesame laser 800 is used for both driving the structure 100 and sensingthe motion of the structure 100, while in certain other embodiments,separate light sources are used for driving the structure 100 andsensing the motion of the structure 100. For example, the light sourceused to drive the structure 100 can have a higher power output (e.g., ina range between about 1 mW to 1 W) than does the light source used tosense the motion of the structure 100. In certain embodiments in whichFP interferometers are used on the drive arms 150, 160 and on the sensearms 180, 190, a narrowband source (e.g., laser) can be used to providelight to each of these FP interferometers. Even in embodiments in whichFP interferometers are not used on the drive arms 150, 160 but are usedon the sense arms 180, 190, a narrowband source (e.g., the same laser asused for sensing) can be used to drive the structure 100. In certainother embodiments in which FP interferometers are not used on the drivearms 150, 160, the gyroscope 10 can comprise two light sources, a source(e.g., having a narrow linewidth or without a narrow linewidth) fordriving the structure 100, and a narrowband source (e.g., laser) forsensing the motion of the structure 100. Such a source without a narrowlinewidth can be advantageously cheaper than a narrowband source incertain such embodiments. The drive arms 150, 160 then oscillate aboutthe drive axis 102 at this torsional resonance frequency. By driving atthe torsional resonance frequency, the structure 100 of certainembodiments responds to a given input energy (e.g., from the radiationpressure) with an oscillation of greater angular amplitude than it wouldat some other frequency. Thus, in certain embodiments as described morefully below, a greater minimum detectable rotation rate isadvantageously achieved.

Another advantage of certain embodiments described herein is that thegyroscope 10 is an active device. The signal strength received from thegyroscope 10 is not only determined passively by the rotation signal,but also actively by how strong the gyroscope 10 is driven by theoperator. Therefore, in contrast to other types of passive sensors(e.g., fiber-optic hydrophones where the signal strength is determinedpassively by only the acoustic signal), in an active device, thesensitivity can be chosen so that the signal falls within the sensitiverange of the device (e.g., driving the gyroscope with an amplitudesufficiently large to operate within a signal range where the signal isstrong but not heavily distorted). This allows in certain embodiments,the detection of both very small signals and very large signals, andincreasing the dynamic range of the active device substantially. Theoperation of an active device by choosing the sensitivity with respectto an initial signal is referred to as closed-loop operation, asschematically shown in FIG. 6. To operate the gyroscope 10 in thisfashion, the rotation signal from the optical sensor system 200 istransmitted to a feedback circuit (negative feedback) to control thedrive system 400 (either electrostatically or optically-driven). Thisnegative feedback can be used in certain embodiments to reduce the driveamplitude when a large rotation signal is present, and similarly toincrease the drive amplitude when a small rotation signal is present. Abenefit of such closed-loop operation in certain embodiments is that itincreases the dynamic range and linearity of the response of the sensorresponse.

The following description provides an example analysis of some of theattributes of an example gyroscope 10 in accordance with certainembodiments described herein. It is a calculation of the minimumdetectable rotation rate for a MEMS structure as shown by FIG. 7 (See H.Ra, W. Piyawattanametha, Y. Taguchi, D. Lee, M. J. Mandella, and O.Solgaard, “Two-dimensional MEMS scanner for dual-axes confocalmicroscopy,” J. Microelectromech. Syst. 16, 969 (2007)), modified toinclude a post 310 (not shown in FIG. 7) in accordance with certainembodiments described herein.

The arm length of the above structure is 1400 μm (total horizontallength 2800 μm). The inner springs are 350 μm long, 8 μm wide, and 30 μmthick. The outer springs are 350 μm long, 8 μm wide, and 60 μm thick,twice the thickness of the inner beams.

The structure 100 has an inner axis (horizontal direction in FIG. 7) andan outer axis (vertical direction in FIG. 7). According to theelectrostatic measurements in Ra et al., the mirror can deflect by 3.6°at resonance (2.9 kHz) for the inner axis, and 6.2° at resonance (500Hz) for the outer axis. The deflection numbers cited in Ra et al. are 4times these numbers (24.8° and 14.4°), because these cited numberscorrespond to the deflection angle of an incident optical beam.

The minimum detectable rotation (MDR) can be calculated for thisstructure with a post attached to the center of the mirror, so thatCoriolis forces can couple the two modes of motion (e.g., about each ofthe inner and outer axes). The calculation described below assumes thatthe inner axis is driven, and the outer axis oscillations are sensed.

The equations of motion for the Foucault pendulum structure have beenanalyzed in other references also (see, e.g., R. T. M'Closkey, S.Gibson, and J. Hui, “System identification of a MEMS gyroscope,” J.Dynamic Syst. Meas. Contr. 123, 201-210 (2001)). It is a basicdamped-spring-mass mechanical model that includes the inertial term, thedamping term, the spring term, and the force term.

The equations of motion for the structure 100 in FIG. 2 are (see, e.g.,R. T. M'Closkey et al.):

$\begin{matrix}{{{Drive}\mspace{14mu} {axis}\text{:}\mspace{14mu} \left\{ {{\left( {I_{\theta} + I_{p}} \right)\frac{\partial^{2}}{\partial t^{2}}} + {\gamma_{\theta}\frac{\partial}{\partial t}} + \kappa_{\theta}} \right\} \theta} = {\tau + {2\; \Omega \; I_{p}\frac{\partial}{\partial t}\varphi}}} & (1) \\{{{Sense}\mspace{14mu} {axis}\text{:}\mspace{14mu} \left\{ {{\left( {I_{\varphi} + I_{p}} \right)\frac{\partial^{2}}{\partial t^{2}}} + {\gamma_{\varphi}\frac{\partial}{\partial t}} + \kappa_{\varphi}} \right\} \varphi} = {{- 2}\; \Omega \; I_{p}\frac{\partial}{\partial t}\theta}} & (2)\end{matrix}$

where I_(θ), I_(φ), and I_(p) denote the moments of inertia of the baseplate 105 around the drive axis, of the base plate 105 around the senseaxis, and of the post 310, respectively. The damping for each axis isdenoted with γ_(θ) and γ_(φ). The torsional spring constants are shownwith κ_(θ) and κ_(φ). The torque applied to the drive axis is τ. Therotation rate applied to the entire structure 100 around the z axis isΩ. A main assumption in using these equations of motion is that themoment of inertia of the post 310 is much larger than the moment ofinertia of the base plate 105, so that the Coriolis coupling is mainlydue to the motion of the post 310. In certain embodiments in which thisis not the case, it is straightforward to include the effect of othermasses in the analysis.

Since θ>>φ, the Coriolis term in Eq. (1) can be neglected. For a drivingtorque τ=τ₀e^(jωt), with ω>>Ω, the angular deflection about the driveaxis from Eq. (1) is θ=θ₀e^(jωt) (disregarding phase shifts), where theangular-deflection amplitude for the inner-axis deflection (drive axis)is:

$\begin{matrix}{{\theta_{0} = {\frac{\tau_{0}}{\kappa_{\theta}}\frac{\omega_{\theta}^{2}}{\sqrt{\left( {\omega^{2} - \omega_{\theta}^{2}} \right)^{2} + {\omega^{2}{\omega_{\theta}^{2}/Q_{\theta}^{2}}}}}}},} & (3)\end{matrix}$

with the resonance frequency

${\omega_{\theta} = \sqrt{\frac{\kappa_{\theta}}{I_{\theta} + I_{p}}}},$

and the quality factor

$Q_{\theta} = {\frac{\kappa_{\theta}}{\gamma_{\theta}\omega_{\theta}} = {\frac{\omega_{\theta}\left( {I_{\theta} + I_{p}} \right)}{\gamma_{\theta}}.}}$

Hence, at ω≈ω_(θ), the drive amplitude is:

$\begin{matrix}{\theta_{0}^{\prime} = {\frac{\tau_{0}Q_{\theta}}{\kappa_{\theta}} = \frac{\tau_{0}}{\gamma_{\theta}\omega_{\theta}}}} & (4)\end{matrix}$

Also, from Eq. (2), the angular deflection about the sense axis isφ=φ₀e^(jωt) (disregarding phase shifts), where the angular-deflectionamplitude for the outer-axis deflection (sense axis) is:

$\begin{matrix}{\varphi_{0} - {\theta_{0}\frac{2\; I_{p}\Omega \; \omega}{\kappa_{\varphi}}\frac{\omega_{\varphi}^{2}}{\sqrt{\left( {\omega^{2} - \omega_{\varphi}^{2}} \right)^{2} + {\omega^{2}{\omega_{\varphi}^{2}/Q_{\varphi}^{2}}}}}}} & (5)\end{matrix}$

where the resonance frequency is:

$\begin{matrix}{\omega_{\varphi} = \sqrt{\frac{\kappa_{\varphi}}{I_{\varphi} + I_{p}}}} & (6)\end{matrix}$

and the Q-factor (in different parameter combinations) is:

$\begin{matrix}{Q_{\varphi} = {\frac{\kappa_{\varphi}}{\gamma_{\varphi}\omega_{\varphi}} = {\frac{\omega_{\varphi}\left( {I_{\varphi} + I_{p}} \right)}{\gamma_{\varphi}} = \frac{\sqrt{\kappa_{\varphi}\left( {I_{\varphi} + I_{p}} \right)}}{\gamma_{\varphi}}}}} & (7)\end{matrix}$

Assuming ω_(φ)>>ω_(θ), i.e., the gyroscope 10 does not sense atresonance and does not get an enhancement by Q_(φ) in the sense axis, atω≈ω_(θ), the sense amplitude is:

$\begin{matrix}{\varphi_{0}^{\prime} = {{\theta_{0}^{\prime}\frac{2\; I_{p}\Omega \; \omega_{\theta}}{\kappa_{\varphi}}} = \frac{2\; \tau_{0}I_{p}\Omega}{\gamma_{\theta}\kappa_{\varphi}}}} & (8)\end{matrix}$

The noise for the outer axis can be calculated by replacing the torqueterm of Eq. (3) with the Johnson-Nyquist equivalent torque τ₀=√{squareroot over (4k_(B)Tγ_(φ)Δf)}.

A. Thermal-Mechanical Noise

The spectral power of the thermal noise about or around the sense axisof the structure 10 is (see, e.g., S. K. Lamoreaux and W. T. Buttler,“Thermal noise limitations to force measurements with torsion pendulums:Applications to the measurement of the Casimir force and its thermalcorrection,” Phys. Rev. E 71, 036109 (2005)):

$\begin{matrix}{{\varphi_{\omega}}^{2} = {\frac{4\; k_{B}T\; \gamma_{\varphi}\Delta \; f}{\kappa_{\varphi}^{2}}\frac{\omega_{\varphi}^{4}}{\left( {\omega^{2} - \omega_{\varphi}^{2}} \right)^{2} + {\omega^{2}{\omega_{\varphi}^{2}/Q_{\varphi}^{2}}}}}} & (9)\end{matrix}$

The noise level is then:

$\begin{matrix}{\varphi_{N} = {\sqrt{{\varphi_{\omega}}^{2}} = {\frac{2\sqrt{k_{B}T\; \gamma_{\varphi}\Delta \; f}}{\kappa_{\varphi}}\frac{\omega_{\varphi}^{2}}{\sqrt{\left( {\omega^{2} - \omega_{\varphi}^{2}} \right)^{2} + {\omega^{2}{\omega_{\varphi}^{2}/Q_{\varphi}^{2}}}}}}}} & (10)\end{matrix}$

The noise level at the drive frequency ω≈ω_(θ) and ω_(φ)>>ω_(θ) is:

$\begin{matrix}{\varphi_{N}^{\prime} = {{\varphi_{N}\left( {\omega \approx \omega_{\theta}} \right)} = \frac{2\sqrt{k_{B}T\; \gamma_{\varphi}\Delta \; f}}{\kappa_{\varphi}}}} & (11)\end{matrix}$

The assumption can be made that the sense axis measurements are limitedby this thermal-noise level given in Eq. (11).

The spring constants can be calculated using the following equationgiven in Ra et al.:

$\begin{matrix}{{\kappa_{\theta} = {\frac{2\; G}{3}{\frac{{tw}^{3}}{l}\left\lbrack {1 - {\frac{192}{\pi^{5}}\frac{w}{t}{\tanh \left( {\frac{\pi}{2}\frac{t}{w}} \right)}}} \right\rbrack}}},{w < t}} & (12)\end{matrix}$

where G is the shear modulus of silicon, and w, t, and l are the width,thickness, and length of the torsional spring, respectively. Thisequation corresponds to the combined spring constant of two equivalenttorsional springs on opposite sides of the structure 100. Calculatingthe spring constant with the parameters aforementioned (also in Ra etal.) yields:

Inner(drive)springs: κ_(θ)=1.52×10⁻⁶(SI units)

Outer(sense)springs: κ_(φ)=3.35×10⁻⁶(SI units)

The measurements in Ra et al. indicate a Q-factor of slightly largerthan 10 for both the inner and outer axes. This Q is lower than mostMEMS gyroscopes (some have Q's of >10000). However, despite this low Q,large angular displacements in the drive axis are possible due to thevertical comb-drive technology employed for the driving mechanism (seeRa et al.). Using a Q of 10, κ_(φ)=3.35×10⁻⁶ (SI units), andω_(φ)=2π×500 Hz, the damping yields using Eq. (7):

$\gamma_{\varphi} = {\frac{\kappa_{\varphi}}{Q_{\varphi}\omega_{\varphi}} = {5.59 \times 10^{- 12}\left( {{SI}\mspace{14mu} {units}} \right)}}$

Therefore, the flat-band noise (Eq. (10)) in a 1-Hz bandwidth, at 300 K(room) temperature yields:

$\varphi_{N}^{\prime} = {\frac{\sqrt{4\; k_{B}T\; \gamma_{\varphi}\Delta \; f}}{\kappa_{\varphi}} = {9.08 \times 10^{- 11}}}$

The vertical deflection on the end of the arm is then (displacementnoise the fiber Fabry-Perot encounters):

L _(N) ′=L _(arm)φ_(N)′=1.27×10⁻¹³ m=1.27×10⁻⁴ nm

where L_(arm)=1400 μm was used. This result is an order of magnitudelarger than what has been previously detected in our earlier acousticsensor work (see O. Kilic, M. Digonnet, G. Kino, and O. Solgaard,“External fibre Fabry-Perot acoustic sensor based on a photonic-crystalmirror,” Meas. Sci. Technol. 18, 3049 (2007)). In comparison, accordingto the calculations in O. Kilic, “Fiber based photonic-crystal acousticsensor,” Ph.D. Thesis, Stanford University (2008), the optoelectronicnoise is in a 1-Hz bandwidth:

L _(N)′=2.25×10⁻⁸ nm(shot-noise limited detection, mirror R=0.99)

L _(N)′=2.07×10⁻⁷ nm(relative intensity noise or RIN limited detection,mirror R=0.99)

L _(N)′=2.36×10⁻⁷ nm(shot-noise limited detection, mirror R=0.90)

L _(N)′=2.17×10⁻⁶ nm(RIN limited detection, mirror R=0.90)

The largest of these numbers is two orders of magnitude smaller than thethermal-noise induced displacement. This calculation shows that evenwhen fiber Fabry-Perot interferometers with R=0.90 mirrors are used, thedominating noise in this structure will be the thermal-mechanical noise.Therefore, it can be assumed that the limiting noise in the exemplarysensor will be the thermal-mechanical noise. In certain embodiments, thestructure 100 is designed such that the noise in Eq. (11) is larger thanthe opto-electronic noise (such as shot noise) and other noise sources.One direct way of achieving this result is by using weak torsionalsprings, e.g., reducing κ_(φ). The MDR can be calculated as describedbelow.

The signal-to-noise ratio (SNR) can be obtained by dividing the signalof Eq. (5), by the noise of Eq. (10):

$\begin{matrix}{{S\; N\; R} = {\frac{\varphi_{0}}{\varphi_{N}} = \frac{\theta_{0}I_{p}\Omega \; \omega}{\sqrt{k_{B}T\; \gamma_{\varphi}\Delta \; f}}}} & (13)\end{matrix}$

For an SNR of 1, the MDR is then:

$\begin{matrix}{\Omega_{\min} = \frac{\sqrt{k_{B}T\; \gamma_{\varphi}\Delta \; f}}{\theta_{0}I_{p}\omega}} & (14)\end{matrix}$

The damping can be replaced with more familiar parameters, so that it iseasier to follow the discussion. Assuming (as described above):

ω≈θ_(θ)  (15)

the MDR (Eq. (13)) yields:

$\begin{matrix}{{S\; N\; R} = {{\theta_{0}^{\prime}\frac{I_{p}\Omega \; \omega_{\varphi}}{\sqrt{k_{B}T\; \gamma_{\varphi}}}} = \frac{\tau_{0}I_{p}\Omega}{\gamma_{\theta}\sqrt{k_{B}T\; \gamma_{\varphi}}}}} & (16)\end{matrix}$

To calculate the MDR in a 1-Hz bandwidth, an example configuration canbe considered in which the post 310 is attached to the center of theMEMS structure 100 and comprises a 2.5-mm long piece of SMF-28® fiber(125 μm diameter, fused silica material), which is compatible with theMEMS dimensions and is also practical to handle.

The moment of inertia of a circular post, rotated around its end,yields:

${I_{p} = {\frac{1}{12}{m\left( {{3\; a^{2}} + {4\; h^{2}}} \right)}}},$

where m is the mass, a the radius, and h is the height. For theparameters above, the value of the moment of inertia of the post 310 is:

I _(p)=1.41×10⁻¹³(SI units)

Adding the post 310 will modify several parameters. The drive frequencyis the resonance frequency of the inner axis. It was originallyω_(θ)=2π×2.9 kHz (see earlier or Ra et al.). The added mass of the post310 will reduce this resonance frequency to:

$\omega_{\theta}^{\prime} = {\sqrt{\frac{\kappa_{\theta}}{I_{\theta} + I_{p}}} = {\sqrt{\frac{\kappa_{\theta}}{\frac{\kappa_{\theta}}{\omega_{\theta}^{2}} + I_{p}}} = {2\; \pi \times 515\mspace{14mu} {Hz}}}}$

Similarly, the outer axis frequency ω_(θ)=2π×500 Hz will be reduced from500 Hz to 2π×420 Hz.

Adding mass to a spring mass system normally increases the Q-factor (seeEq. (7)), since the damping is not affected much by the additional areaof the mass. However, a worst case scenario assumption can be made thatthe Q-factor does not increase for the case of an added post 310.Therefore, it can be assumed that the inner (drive) axis at 3.6° can bedriven at resonance. Even if the Q-factor would increase, the comb-driveactuators can be provided with a smaller electrical power, so that theoscillation amplitude does not increase beyond this number. Now the MDRcan be calculated. From Eq. (14):

$\Omega_{\min} = {\frac{1}{\theta_{0}I_{p}\omega}\sqrt{\frac{k_{B}T\; \kappa_{\varphi}\Delta \; f}{Q_{\varphi}\omega_{\varphi}}}}$

For the operating frequency ω=ω_(θ)′=2π×515 Hz, and the parametersω_(φ)′=2π×420 Hz, θ₀=3.6°×π/180°, Q_(φ)=10, I_(p)=1.41×10⁻¹³ (SI units),κ_(φ)=3.35×10⁻⁶ (SI units), and Δf=1 Hz, the MDR is:

Ω_(min)=25.3×10⁻⁶ rad/s=25.3 μrad/s≈5°/h

The values used in this example were not selected to yield the bestpossible MDR. With straightforward, practical improvements in thesedesign features, in particular a two-fold increase in the driveamplitude and in the post mass, a theoretical MDR of ˜1°/h is entirelypossible.

In certain embodiments as described above, oscillations or vibrations ofthe structure 100 around the drive axis can be driven by utilizingradiation pressure. While it is more difficult to obtain large driveamplitudes with optical actuation, as compared to electrostaticactuation, it can be useful for certain applications (such aselectromagnetically harsh environments). For a Fabry-Perot cavity withfinesse F, addressed with an incident optical power of P_(in), the forcedue to radiation pressure is:

$\begin{matrix}{F_{RP} = {\frac{2\; P_{in}}{c}\frac{F}{\pi}}} & (17)\end{matrix}$

This expression is only valid for small displacements, such that theresonance wavelength of the cavity is not changed much compared to itslinewidth. Therefore, in certain embodiments, the dynamic range for ahigh finesse cavity will be limited.

The applied torque is:

$\begin{matrix}{{\tau_{0} = {{F_{RP}L_{arm}} = {\frac{2\; F}{\pi \; c}P_{in}L_{arm}}}},} & (18)\end{matrix}$

where L_(arm) is the length of the base arm. The equation to beoptimized, with all relevant parameters, is then:

$\begin{matrix}{{\Omega_{\min} = {C\frac{\sqrt{\gamma_{\theta}^{2}\gamma_{\varphi}\Delta \; f}}{{FP}_{in}L_{arm}I_{p}}}},} & (19)\end{matrix}$

where C is a constant equal to

$C = {\frac{1}{2}\pi \; c{\sqrt{k_{B}T}.}}$

Hence, for a smaller minimum detectable rotation, F P_(in)L_(arm)I_(p)in certain embodiments is increased, and the damping γ decreased.

Example Method to Accurately Find the Center of the Mirror

In certain embodiments, when the post structure 310 is not well centeredon the base plate 105, the drive oscillation is not purely a torsionalmode, but has a component that also moves the structure 100 up and down.This motion is usually referred to as a rocking mode. Although thisrocking mode of the structure 100 can be filtered out by thedifferential sensing described herein, it may still reduce the dynamicrange by creating a distortion in the rotation signal. To reduce oreliminate the rocking mode in certain embodiments, the post structure310 can to be centered on the base plate 105 more accurately. Below, anexample method to find the center of the mirror is described inaccordance with certain embodiments described herein.

The fiber Fabry-Perot displacement sensors used to sense theoscillations in certain embodiments of the gyroscope are very sensitive.In certain embodiments, such sensors allow the detection of the Brownianmotion of the mirror, referred to as the thermal-mechanical noise. Whena fiber sensor is placed to be in optical communication with a portionof the sense axis, the noise spectrum detected by the fiber sensor willhave a peak at the resonance frequency of the sense mode. Similarly,when a fiber sensor is placed to be in optical communication with aportion of the drive axis, the noise spectrum detected by the fibersensor will have a peak at the resonance frequency of the drive mode. Incertain embodiments, the closer to the center of the structure 100 thefiber sensor is placed, the smaller will be these peaks in the detectednoise spectra. By moving the fiber sensor towards a position where thesepeaks are reduced to the smallest value, it is possible to find thecenter of the structure 100 as accurately as possible for certainembodiments. Once this center position is determined, it can either bemarked (e.g., using a very small amount of paint at the tip of the fibersensor), or the fiber sensor itself can be fixed to that position (e.g.,by a small amount of epoxy) and then cut or otherwise sized so that itcan serve as a post structure 310.

Example Methods of Detecting Rotation

FIG. 8 is a flowchart of an example method 2000 of detecting rotation inaccordance with certain embodiments described herein. The method 2000comprises providing a structure 100 configured to be driven to moveabout a drive axis 102 and to move about a sense axis 104 in response toa Coriolis force 1000 generated by rotation of the structure 100 about arotational axis z while moving about the drive axis 102, as shown inoperational block 2010 of FIG. 8. The method 2000 also comprises drivingthe structure 100 to move about the drive axis 102, as shown inoperational block 2020; and rotating the structure 100 about therotational axis z while the structure 100 moves about the drive axis102, as shown in operational block 2030. The method 2000 furthercomprises optically measuring movement of the structure 100 about thesense axis 104, as shown in operational block 2040 of FIG. 8.

In certain embodiments, the method 2000 is compatible with variousconfigurations of the gyroscope 10 described herein. For example, themethod 2000 is compatible with a gyroscope 10 comprising a structure 100which comprises a generally planar portion 105, at least one firsttorsional spring 110 extending generally along the drive axis 102 andoperationally coupling the generally planar portion 105 to a supportstructure 300, and at least one second torsional spring 120 extendinggenerally along the sense axis 104 and operationally coupling thegenerally planar portion 105 to the support structure 300.

In certain embodiments, the generally planar portion 105 comprises atleast two drive arms 150, 160 extending in opposite directions from oneanother generally along the sense axis 104. The second torsional spring120 operationally couples the two drive arms 150, 160 to the supportstructure 300.

In certain embodiments, the generally planar portion 105 comprises atleast two sense arms 180, 190 extending in opposite directions from oneanother generally along the drive axis 102. The first torsional spring110 operationally couples the two sense arms 180, 190 to the supportstructure 300.

In certain embodiments, the structure 100 comprises a post portion 310extending generally perpendicularly away from the generally planarportion 105. In certain embodiments, the structure 100 (including one ormore of the generally planar portion 105 and the post portion 310) ismicromachined and comprises silicon, polysilicon, silica, or quartz. Incertain embodiments, the post portion 310 comprises a short length(e.g., 10-20 millimeters) of a conventional optical fiber (e.g., withits jacket stripped off).

In certain embodiments, driving the structure 100 in the operationalblock 2020 comprises irradiating at least a portion 430 of the structure100 with electromagnetic radiation having sufficient radiation pressureto drive the structure 100 to oscillate about the drive axis 102 (e.g.,with an amplitude large enough to achieve a desired level of sensitivityof at least 0.0015 degree/hour, or across a range between 0.0015degree/hour and 15 degrees/hour, or across a range between 0.0015degree/hour and 1.296×10⁷ degrees/hour). In certain other embodiments,driving the structure 100 in the operational block 2020 comprisesapplying sufficient electrostatic force on at least a portion 480 of thestructure to drive the structure 100 to oscillate about the drive axis102 (e.g., with an amplitude large enough to achieve a desired level ofsensitivity).

In certain embodiments, the drive axis 102 is substantially planar withand substantially perpendicular to the sense axis 104. In addition, incertain embodiments, the rotational axis z is substantiallyperpendicular to at least one of the drive axis 102 and the sense axis104. In certain embodiments, optically measuring movement in theoperational block 2040 comprises irradiating at least a portion 630 ofthe structure 100 with electromagnetic radiation, as shown inoperational block 2041 in FIG. 9, and receiving reflectedelectromagnetic radiation from the portion 630 of the structure 100, asshown in operational block 2042 in FIG. 9. In certain embodiments,optically measuring movement in the operational block 2040 furthercomprises detecting at least a portion of the received reflectedelectromagnetic radiation as shown in operational block 2043 in FIG. 9,and generating one or more signals in response to the detected portionof the received reflected electromagnetic radiation, as shown inoperational block 2044 in FIG. 9.

Various embodiments have been described above. Although this inventionhas been described with reference to these specific embodiments, thedescriptions are intended to be illustrative of the invention and arenot intended to be limiting. Various modifications and applications mayoccur to those skilled in the art without departing from the true spiritand scope of the invention as defined in the appended claims.

1. A gyroscope comprising: a structure configured to be driven to moveabout a drive axis, the structure further configured to move about asense axis in response to a Coriolis force generated by rotation of thestructure about a rotational axis while moving about the drive axis; atleast one first torsional spring extending generally along the driveaxis and operationally coupling the structure to a support structure; atleast one second torsional spring extending generally along the senseaxis and operationally coupling the structure to the support structure;and an optical sensor system configured to optically measure movement ofthe structure about the sense axis.
 2. The gyroscope of claim 1, whereinthe structure comprises at least two portions extending in oppositedirections from one another generally along the sense axis, wherein theat least one second drive torsional spring operationally couples the atleast two portions to the support structure.
 3. The gyroscope of claim1, wherein the structure comprises at least two portions extending inopposite directions from one another generally along the drive axis,wherein the at least one first torsional spring operationally couplesthe at least two portions to the support structure.
 4. The gyroscope ofclaim 1, wherein the structure comprises a base portion and an elongateportion extending generally perpendicularly away from the base portion.5. The gyroscope of claim 1, wherein the rotational axis issubstantially perpendicular to at least one of the drive axis and thesense axis.
 6. The gyroscope of claim 1, wherein the optical sensorsystem comprises one or more optical waveguides configured to irradiateat least a portion of the structure with electromagnetic radiation andto receive reflected electromagnetic radiation from the portion of thestructure.
 7. The gyroscope of claim 6, wherein the one or more opticalwaveguides and the portion of the structure form at least oneFabry-Perot cavity therebetween.
 8. The gyroscope of claim 6, whereinthe portion of the structure comprises one or more photonic-crystalstructures.
 9. The gyroscope of claim 1, further comprising a drivesystem configured to drive the structure to oscillate about the driveaxis.
 10. A method of detecting rotation, the method comprising:providing a structure configured to be driven to move about a drive axisand to move about a sense axis in response to a Coriolis force generatedby rotation of the structure about a rotational axis while moving aboutthe drive axis, wherein the structure is operatively coupled to asupport structure by at least one first torsional spring and at leastone second torsional spring, the at least one first torsional springextending generally along the drive axis and at least one secondtorsional spring extending generally along the sense axis; driving thestructure to move about the drive axis; rotating the structure about therotational axis while the structure moves about the drive axis; andoptically measuring movement of the structure about the sense axis. 11.The method of claim 10, wherein the structure comprises a base portionand an elongate portion extending generally perpendicularly away fromthe base portion.
 12. The method of claim 10, wherein driving thestructure comprises irradiating at least a portion of the structure withelectromagnetic radiation having sufficient radiation pressure to drivethe structure to oscillate about the drive axis.
 13. The method of claim10, wherein driving the structure comprises applying sufficientelectrostatic force on at least a portion of the structure to drive thestructure to oscillate about the drive axis.
 14. The method of claim 10,wherein optically measuring movement of the structure further comprisesdetecting at least a portion of the received reflected electromagneticradiation and generating one or more signals in response to the detectedportion of the received reflected electromagnetic radiation.
 15. Amethod of fabricating a gyroscope, the method comprising operationallycoupling a movable structure to a support structure by at least onefirst torsional spring and by at least one second torsional spring,wherein the movable structure is configured to move about a sense axisin response to a Coriolis force generated by rotation of the movablestructure about a rotational axis while the movable structure movesabout a drive axis, wherein the at least one first torsional springextends generally along the drive axis and the at least one secondtorsional spring extends generally along the sense axis.
 16. The methodof claim 15, further comprising positioning an optical sensor systemconfigured to optically measure movement of the movable structure aboutthe sense axis.
 17. The method of claim 16, wherein the optical sensorsystem is configured to irradiate at least a portion of the movablestructure with electromagnetic radiation in a direction of the movementabout the sense axis and to receive reflected electromagnetic radiationfrom the portion of the movable structure.
 18. The method of claim 15,wherein the movable structure comprises a base portion and an elongateportion extending generally perpendicularly away from the base portion,the method further comprising centering the elongate portion on themovable structure.
 19. The method of claim 18, wherein centering theelongate portion comprises: measuring a plurality of firstthermal-mechanical noise spectra of the movable structure at acorresponding plurality of positions on the sense axis, each firstthermal-mechanical noise spectrum having a first peak at a resonancefrequency of a sense mode; measuring a plurality of secondthermal-mechanical noise spectra of the movable structure at acorresponding plurality of positions on the drive axis, each secondthermal-mechanical noise spectrum having a second peak at a resonancefrequency of a drive mode; and determining a position of the elongateportion on the movable structure where the first and second peaks arereduced.